Properties of Search Algorithms:
Following are the four essential properties of search algorithms to
compare the efficiency of these algorithms:
Completeness: A search algorithm is said to be complete if it
guarantees to return a solution if at least any solution exists for any
random input.
Optimality: If a solution found for an algorithm is guaranteed to be
the best solution (lowest path cost) among all other solutions, then
such a solution for is said to be an optimal solution.
Time Complexity: Time complexity is a measure of time for an
algorithm to complete its task.
Space Complexity: It is the maximum storage space required at any
point during the search, as the complexity of the problem.
UNIT-II
Uninformed Searching strategies-Breadth First Search, Depth First
search, Depth lim ited search, Iterative deepening search,
Bidirectional Search - Avoiding repeated States - Searching with
Partial information Informed search strategies Greedy Best First
Search-A* Search-Heuristic Functions Local Search Algorithms for
Optimization Problems-Local search in Continuous Spaces
Types of search algorithms
Based on the search problems we can classify the search algorithms
into uninformed (Blind search) search and informed search
(Heuristic search) algorithms.
Uninformed/Blind Search:
The uninformed search does not contain any domain knowledge such
as closeness, the location of the goal. It operates in a brute-force way
as it only includes information about how to traverse the tree and
how to identify leaf and goal nodes. Uninformed search applies a way
in which search tree is searched without any information about the
search space like initial state operators and test for the goal, so it is
also called blind search. It examines each node of the tree until it
achieves the goal node.
It can be divided into five main types:
o Breadth-first search
o Uniform cost search
o Depth-first search
o Iterative deepening depth-first search
o Bidirectional Search
Informed Search
Informed search algorithms use domain knowledge. In an informed
search, problem information is available which can guide the search.
Informed search strategies can find a solution more efficiently than
an uninformed search strategy. Informed search is also called a
Heuristic search.
A heuristic is a way which might not always be guaranteed for best
solutions but guaranteed to find a good solution in reasonable time.
Informed search can solve much complex problem which could not
be solved in another way.
An example of informed search algorithms is a traveling salesman
problem.
1. Greedy Search
2. A* Search
Uninformed Search Algorithms
Uninformed search is a class of general-purpose search algorithms
which operates in brute force-way.
Uninformed search algorithms do not have additional information
about state or search space other than how to traverse the tree, so it
is also called blind search.
Following are the various types of uninformed search algorithms:
1. Breadth-first Search
2. Depth-first Search
3. Depth-limited Search
4. Iterative deepening depth-first search
5. Uniform cost search
6. Bidirectional Search
1. Breadth-first Search:
o Breadth-first search is the most common search strategy for
traversing a tree or graph. This algorithm searches breadthwise
in a tree or graph, so it is called breadth-first search.
o BFS algorithm starts searching from the root node of the tree
and expands all successor node at the current level before
moving to nodes of next level.
o The breadth-first search algorithm is an example of a general-
graph search algorithm.
o Breadth-first search implemented using FIFO queue data
structure.
Advantages:
o BFS will provide a solution if any solution exists.
o If there are more than one solutions for a given problem, then
BFS will provide the minimal solution which requires the least
number of steps.
Disadvantages:
o It requires lots of memory since each level of the tree
saved into memory to expand the next level.
must be
o BFS needs lots of time if the solution is far away from the root
node.
Example:
In the below tree structure, we have shown the traversing of the tree
using BFS algorithm from the root node S to goal node K. BFS search
algorithm traverse in layers, so it will follow the path which is shown
by the dotted arrow, and the traversed path will be:
1. S---> A--->B---->C--->D---->G--->H--->E---->F---->I--- >K
Time Complexity:
Time Complexity of BFS algorithm can be obtained by the number of
the d=
nodes traversed in BFS until the shallowest Node. Where
depth of shallowest solution and b is a node at every state.
T (b) = 1+b
2
+b
3
+.......+ b
d
= O (b
d
)
Space Complexity: Space complexity of BFS algorithm is given by the
Memory size of frontier which is O(b
d
).
Completeness: BFS is complete, which means if the shallowest goal
node is at some finite depth, then BFS will find a solution.
Optimality: BFS is optimal if path cost is a non-decreasing function
of the depth of the node.
2. Depth-first Search
o Depth-first search isa recursive algorithm for traversing a tree
or graph data structure.
o It is called the depth-first search because it starts from the root
node and follows each path to its greatest depth node before
moving to the next path.
o DFS uses a stack data structure for its implementation.
o The process of the DFS algorithm is sim ilar to the BFS
algorithm.
Advantage:
o DFS requires very less memory as it only needs to store a stack
of the nodes on the path from root node to the current node.
o It takes less time to reach to the goal node than BFS algorithm
(if it traverses in the right path).
Disadvantage:
o There is the possibility that many states keep re-occurring, and
there is no guarantee of finding the solution.
o DFS algorithm goes for deep down searching and sometime it
may go to the infinite loop.
Example:
In the below search tree, we have shown the flow of
search, and it will follow the order as:
depth-first
Root node--->Left node ----> right node.
It will start searching from root node S, and traverse A, then B, then
D a n d E, after traversing E, it will backtra ck the tree as E has no
other successor and still goal node is not found. After backtracking it
will traverse n od e C a nd then G, an d he re it will termina te as it
f oun d goal node .
Completeness: DFS search algorithm is complete within finite state
space as it will expand every node within a limited search tree.
Time Complexity: Time complexity of DFS will be equivalent to the
node traversed by the algorithm.
It is given by:
T(n)= 1+ n
2
+ n
3
+.........+ n
m
=O(n
m
)
Where, m= maximum depth of any node and this can be much larger
than d (Shallowest solution depth)
Space Complexity: DFS algorithm needs to store only single path
from the root node, hence space complexity of DFS is equivalent to
the size of the fringe set, which is O(bm).
Optimal: DFS search algorithm is non-optimal, as it may generate a
large number of steps or high cost to reach to the goal node.
3. Depth-Limited Search Algorithm:
A depth-limited search algorithm is similar to depth-first search with
a predetermined limit. Depth-limited search can solve the drawback
of the infinite path in the Depth-first search. In this algorithm, the
node at the depth limit will treat as it has no successor nodes
further.
Depth-limited search can be terminated with two Conditions of
failure:
o Standard failure value: It indicates that problem does not have
any solution.
o Cutoff failure value: It defines no solution for the problem
within a given depth limit.
Advantages:
Depth-limited search is Memory efficient.
Disadvantages:
o Depth-limited search also has a disadvantage of
incompleteness.
o It may not be optimal if the problem has more than one
solution.
Example:
Completeness: DLS search algorithm is complete if the solution is
above the depth-limit.
Time Complexity: Time complexity of DLS algorithm is O(b ).
Space Com plexity: Space complexity of DLS algorithm is O .
Optimal: Depth-limited search can be viewed as a special case of
4. Uniform-cost Search Algorithm:
Uniform-cost search is a searching algorithm used for traversing a
weighted tree or graph. This algorithm comes into play when a
different cost is available for each edge. The primary goal of the
uniform-cost search is to find a path to the goal node which has the
lowest cumulative cost. Uniform-cost search expands nodes
according to their path costs form the root node. It can be used to
solve any graph/tree where the optimal cost is in demand. A uniform-
cost search algorithm is implemented by the priority queue. It gives
maximum priority to the lowest cumulative cost. Uniform cost search
is equivalent to BFS algorithm if the path cost of all edges is the
same.
Advantages:
o Uniform cost search is optimal because at every state the path
with the least cost is chosen.
Disadvantages:
o It does not care about the number of steps involve in searching
and only concerned about path cost. Due to which this
algorithm may be stuck in an infinite loop.
Example:
Completeness:
Uniform-cost search is complete, such as if there is a solution,
UCS will find it.
Time Complexity:
Let C* is Cost of the optimal solution, and
is each step to get
Hence, the worst-case time complexity of Uniform-cost search isO(b
1
)/.
Space Complexity:
The same logic is for space complexity so, the worst-case space
complexity of Uniform-cost search is O(b ).
Optimal:
Uniform-cost search is always optimal as it only selects a path
with the lowest path cost.
5. Iterative deepening depth-first Search:
The iterative deepening algorithm is a combination of DFS and BFS
algorithms. This search algorithm finds out the best depth limit and
does it by gradually increasing the limit until a goal is found.
This algorithm performs depth-first search up to a certain "depth
limit", and it keeps increasing the depth limit after each iteration
until the goal node is found.
This Search algorithm combines the benefits of Breadth-first search's
fast search and depth-first search's memory efficiency.
The iterative search algorithm is useful uninformed search when
search space is large, and depth of goal node is unknown.
Advantages:
o It combines the benefits of BFS and DFS search algorithm in
terms of fast search and memory efficiency.
Disadvantages:
o The main drawback of IDDFS is that it repeats all the work of
the previous phase.
Example:
Following tree structure is showing the iterative deepening depth-
first search. IDDFS algorithm performs various iterations until it
does not find the goal node. The iteration performed by the algorithm
is given as:
1'stIteration ----->A
2'ndIteration-- >A,B,C
3'rdIteration----- >A,B,D,E,C,F,G
4'thIteration ---->A,B,D,H,I,E,C,F,K,G
In the fourth iteration, the algorithm will find the goal node.
Completeness:
This algorithm is complete is if the branching factor is finite.
Time Complexity:
Let's suppose b is the branching factor and depth is d then the
worst-case time complexity is O(b
d
).
Space Complexity:
The space complexity of IDDFS will be O(bd).
Optimal:
IDDFS algorithm is optimal if path cost is a non- decreasing function
of the depth of the node.
6. Bidirectional Search Algorithm:
Bidirectional search algorithm runs two simultaneous searches, one
form initial state called as forward-search and other from goal node
called as backward-search, to find the goal node. Bidirectional search
replaces one single search graph with two small subgraphs in which
one starts the search from an initial vertex and other starts from goal
vertex. The search stops when these two graphs intersect each
other.Bidirectional search can use search techniques such as BFS,
DFS, DLS, etc.
Advantages:
o Bidirectional search is fast.
o Bidirectional search requires less memory
Disadvantages:
o Implementation of the bidirectional search tree is difficult.
o In bidirectional search, one should know the goal state in
advance.
Example:
In the below search tree, bidirectional search algorithm is applied.
This algorithm divides one graph/tree into two sub-graphs. It starts
traversing from node 1 in the forward direction and starts from goal
node 16 in the backward direction. The algorithm terminates at node
9 where two searches meet.
Completeness: Bidirectional Search is complete if we use BFS in both
searches.
Time Complexity: Time complexity of bidirectional search using BFS
is O(b
d
).
Space Complexity: Space complexity of bidirectional search is O(b
d
).
Optimal: Bidirectional search is Optimal.
Informed Search Algorithms
So far we have talked about the uninformed search algorithms which
looked through search space for all possible solutions of the problem
without having any additional knowledge about search space. But
informed search algorithm contains an array of knowledge such as
how far we are from the goal, path cost, how to reach to goal node,
etc. This knowledge helps agents to explore less to the search space
and find more efficiently the goal node.
The informed search algorithm is more useful for large search space.
Informed search algorithm uses the idea of heuristic, so it is also
called Heuristic search.
Heuristics function: Heuristic is a function which is used in Informed
Search, and it finds the most promising path. It takes the current
state of the agent as its input and produces the estimation of how
close agent is from the goal.
The heuristic method, however, might not always give the best
solution, but it guaranteed to find a good solution in reasonable time.
Heuristic function estimates how close a state is to the goal. It is
represented by h(n), and it calculates the cost of an optimal path
between the pair of states. The value of the heuristic function is
always positive.
Admissibility of the heuristic function is given as: h(n) <= h*(n)
Here h(n) is heuristic cost, and h*(n) is the estimated cost.
Hence heuristic cost should be less than or equal to the estimated
cost.
Pure Heuristic Search:
Pure heuristic search is the simplest form of heuristic search
algorithms. It expands nodes based on their heuristic value h(n). It
maintains two lists, OPEN and CLOSED list. In the CLOSED list, it
places those nodes which have already expanded and in the OPEN
list, it places nodes which have yet not been expanded.
On each iteration, each node n with the lowest heuristic value is
expanded and generates all its successors and n is placed to the
closed list. The algorithm continues unit a goal state is found.
In the informed search we will discuss two main algorithms which
are given below:
o Best First Search Algorithm(Greedy search)
o A* Search Algorithm
1.) Best-first Search Algorithm (Greedy Search):
Greedy best-first search algorithm always selects the path which
appears best at that moment. It is the combination of depth-first
search and breadth-first search algorithms. It uses the heuristic
function and search. Best-first search allows us to take the
advantages of both algorithms. With the help of best-first search, at
each step, we can choose the most promising node. In the best first
search algorithm, we expand the node which is closest to the goal
node and the closest cost is estimated by heuristic function, i.e.
f(n)= g(n).
Were, h(n)= estimated cost from node n to the goal.
The greedy best first algorithm is implemented by the priority queue.
Best first search algorithm:
o Step 1: Place the starting node into the OPEN list.
o Step 2: If the O P E N list is empty, Stop and return failure.
o Step 3: Remove the node n, from the OPEN list which has the
lowest value of h(n), and places it in the CLOSED list.
o Step 4: Expand the node n, and generate the successors of node
n.
o Step 5: Check each successor of node n, and find whether any
node is a goal node or not. If any successor node is goal node,
then return success and terminate the search, else proceed to
Step 6.
o Step 6: For each successor node, algorithm checks for
evaluation function f(n), and then check if the node has been in
either OPEN or CLOSED list. If the node has not been in both
list, then add it to the OPEN list.
o Step 7: Return to Ste p 2.
Advantages:
o Best first search can switch between BFS and DFS by gaining
the advantages of both the algorithms.
o This algorithm is more efficient than BFS and DFS algorithms.
Disadvantages:
o It can behave as an unguided depth-first search in the worst
case scenario.
o It can get stuck in a loop as DFS.
o This algorithm is not optimal.
Example:
Consider the below search problem, and we will traverse it using
greedy best-first search. At each iteration, each node is expanded
which is given in the below
using evaluation function f(n)=h(n) ,
table.
In this search exam ple, we
are OPEN and CLOSED Lists.
traversing the above example.
are using two
Following are the
lists which
iteration for
Expand the nodes of S and put in the CLOSED list
Initialization: Open [A, B], Closed [S]
Iteration 1: Open [A], Closed [S, B]
Iteration2: Open[E,F,A],Closed[S,B]
: Open [E, A], Closed [S, B, F]
Iteration3: Open[I,G,E,A],Closed[S,B,F]
: Open [I, E, A], Closed [S, B, F, G]
Hence the final solution path will be: S----> B----->F----> G
Time Complexity: The worst case time complexity of Greedy best first
search is O(b
m
).
Space Complexity: The worst case space complexity of Greedy best
first search is O(b
m
). Where, m is the maximum depth of the search
space.
Complete: Greedy best-first search is also incomplete, even if the
given state space is finite.
Optimal: Greedy best first search algorithm is not optimal.
2.) A* Search Algorithm:
A* search is the most commonly known form of best-first search. It
uses heuristic function h(n), and cost to reach the node n from the
start state g(n). It has combined features of UCS and greedy best-first
search, by which it solve the problem efficiently. A* search algorithm
finds the shortest path through the search space using the heuristic
function. This search algorithm expands less search tree and
provides optimal result faster.
A* algorithm is similar to UCS except that it uses g(n)+h(n) instead of
g(n).
In A* search algorithm, we use search heuristic as well as the cost to
reach the node.
Hence we can combine both costs as following, and this sum is called
as a fitness number.
Algorithm of A* search:
Step1: Place the starting node in the OPEN list.
Step 2: Check if the OPEN list is empty or not, if the list is empty
then return failure and stops.
Step 3: Select the node from the OPEN list which has the smallest
value of evaluation function (g+h), if node n is goal node then return
success and stop, otherwise
Step 4: Expand node n and generate all of its successors, and put n
into the closed list. For each successor n', check whether n' is
already in the OPEN or CLOSED list, if not then compute evaluation
function for n' and place into Open list.
Step 5: Else if node n' is already in OPEN and CLOSED, then it
should be attached to the back pointer which reflects the lowest g(n')
value.
Step 6: Return to Step 2.
Advantages:
o A* search algorithm is the best algorithm than other search
algorithms.
o A* search algorithm is optimal and complete.
o This algorithm can solve very complex problems.
Disadvantages:
o It does not always produce the shortest path as it mostly based
on heuristics and approximation.
o A* search algorithm has some complexity issues.
o The main drawback of A* is memory requirement as it keeps all
generated nodes in the memory, so it is not practical for various
large-scale problems.
Example:
In this example, we will traverse the given graph using the A*
algorithm. The heuristic value of all states is given in the below table
so we will calculate the f(n) of each state using the formula f(n)= g(n)
+ h(n), where g(n) is the cost to reach any node from start state.
Here we will use OPEN and CLOSED list.
Solution :
Initialization: {(S, 5)}
Iteration1: {(S--> A, 4), (S-->G, 10)}
Iteration2: {(S--> A-->C, 4), (S--> A-->B, 7), (S-->G, 10)}
Iteration3: {(S--> A-->C--->G, 6), (S--> A-->C--->D, 11), (S--> A-->B,
7), (S-->G, 10)}
Iteration 4 will give the final result, as S--->A--->C--->G it provides
the optimal path with cost 6.
Points to remember:
o A* algorithm returns the path which occurred first, and it does
not search for all remaining paths.
o The efficiency of A* algorithm depends on the quality of
heuristic.
o A* algorithm expands all nodes which satisfy the condition
f(n)<="" li="">
Complete: A* algorithm is complete as long as:
o Branching factor is finite.
o Cost at every action is fixed.
Optimal: A* search algorithm is optimal if it follows below two
conditions:
o Admissible: the first condition requires for optimality is that
h(n) should be an admissible heuristic for A* tree search. An
admissible heuristic is optimistic in nature.
o Consistency: Second required condition is consistency for only
A* graph-search.
If the heuristic function is admissible, then A* tree search will always
find the least cost path.
Time Complexity: The time complexity of A* search algorithm
depends on heuristic function, and the number of nodes expanded is
exponential to the depth of solution d. So the time complexity is
O(b^d), where b is the branching factor.
Space Complexity: The space complexity of A* search algorithm
is O(b^d)
